Aah, the sound of shaking animal intestines.. I
mean, strings which are traditionally made out of cat gut but regardless of
what it's made out of when a string vibrates it does so with the ends fixed to the
instrument. This means that it can only vibrate in certain waves, sin waves. Like a
jump rope with one bump or two bumps or three or four or some combination of
these bumps. The more bumps the higher the pitch and the faster the string has to
vibrate. In fact, the frequency of a strings vibration is exactly equal to the
number of bumps times the strings fundamental frequency that is, the
frequency of vibrations for a single bump. And since most melodious instruments use
either strings or air vibrating pipes which has the same sinusoidal
behavior it won't surprise you to hear that musicians have different names for
the different ratios between these pitches. In the traditional Western scale, 1 to 2
bumps is called an octave; 2 to 3 is a perfect fifth; 3 to 4 is a perfect fourth, then a major third, minor third some other things that aren't on the scale and from 8 to 9 bumps is a
major second or whole step. If you play a few of these notes together you get the nice sound of
perfect harmony. Hence the name for this band of pitches, harmonics. In fact a sound that matches one
of the harmonics of a string can cause that string to start vibrating on its
own with their resonant ringing sound. And a bugle playing taps uses only the notes in a single series of harmonics which
is part of why the melody of taps rings so purely and why you can play taps with the
harmonics of a single guitar string. Harmonics can also be used to tune
string instruments. For example, on a violin, viola or cello, the third harmonic
on one string should be equal to the second harmonic on the next string up. Bassists and
guitarists can compare the fourth harmonic to the third harmonic on
the next string up but then we come to the piano or historically the harpsichord or
clavichord but either way the problem is this: it has too many strings. There's a
string for each of the 12 semi tones of the Western scale times seven. If you
wanted to tune these strings using harmonics you could for example try
using whole steps that is you could compare the ninth harmonic on one key to the eighth
harmonic two keys up which works fine for the first few keys; but if you do it six times, you'll get to what's supposed
to be the original note an octave up which should have twice the frequency. Except that our harmonic tuning method
multiplied the frequency by a factor of nine eighths each time and 9 over 8 to the 6th is not two, its 2.027286529541 etcetera. If you tried harmonically tuning a piano using major
thirds instead, you'd multiply the frequency by five fourths three times or
1.953125, still not two. Using fourths you'd get 1.973 not two. Fifths
gives 2.027 again. And don't even try using half steps; you will be off by almost 10
percent and this is the problem. It's mathematically impossible to tune a
piano consistently across all keys using perfect beautiful harmonics, so we don't.
Most pianos these days use what's called equal tempered tuning where the
frequency of each key is the 12th root of two times the frequency of the key below it.
The 12th root of 2 is an irrational number something you never get using simple ratios of harmonic tuning; but its
benefit is that once you go up 12 keys you end up with exactly the 12th root of 2 to the 12th or,
twice the frequency. Perfect octave! However, the octave is the only perfect
interval on an equally tuned piano. Fifths are slightly flat; fourths are slightly sharp;
major thirds are sharp, minor thirds are flat and so on. You can hear a kind of "wawawawawa" effect in this equal tempered chord; which goes away
using harmonic tune. But, if you tuned an instrument using the 12th root of 2 as most pianos, digital tuners and computer instruments are, you can play any song, in any key, and they will all be
equally and just slightly out of tune. This Minute Physics video is brought to
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This is also one of the things that makes a big difference in playing a non-fixed-pitch instrument at a high level. To make just intonation work, you're basically always beholden to a reference pitch.
So let's say I'm playing trumpet and I'm holding out a C while the ensemble around me is playing a C major chord. Great, so I just make a perfect octave with the lowest bass voice and I'm good... but then the ensemble moves to an A minor chord and I'm still holing the same note. Well, now I'm not playing the root, I'm on the minor 3rd of a chord. Minor thirds are just a little bit sharp in equal temperament by something like 12 cents (I no longer remember the numbers exactly because my ear does it all). So now I have to quickly make a very subtle adjustment to the note that I'm already holding to be in tune with the ensemble.
This can be infuriating with mid-tier players as their pitch can be wonky. If I'm on the 3rd and the people around me are making the 5th too narrow, it's hard for me to find the absolute center of that 3rd. In fact, this actually makes me very tired playing trumpet because my lip works extra hard trying to compensate but never can really find the center. But the better people you're playing with, the more pitch aware they are. They have developed good enough ears to quickly adjust depending if they are on a major 3rd, minor 3rd, 5th, 7th, or whatever to be constantly in tune. It's much the same way a barbershop quartet or other a capella vocal group maintains such an absolute pure sound. Everything is retuning with reference to the root of each harmony.
Actually, you shouldn't tune a guitar with harmonics for this same reason.
it doesn't even address the nearly as large issue of octave stretching.
This is a crash course if there ever was one...
And now on to a weekend of math and acoustics on wikipedia.
This is also why old composers claimed that each key had its own characteristics.
In equal temperament, all twelve keys sound the same. The only real difference is in the tone quality of whatever register of the instrument you're playing.
However, up until the early to mid 20th century, most pianos were tuned to some form of well-tempered system (one of the more popular ones is Werckmeister III): tuning systems that were intervallic, but slightly adjusted so as to avoid "wolf tones" in certain keys. These keyboards could play in any key and not sound horribly out of tune (and many keys sounded better in this type of tuning than in equal temperament), but each key would have different intervals between each step, so each key would have a slightly different quality than the others.
Equal temperament only became popular with the rise of atonality, where equal intervals between each half step became more important than harmonic consonance.
Ok... now what about woodwinds like saxes? How does that figure into all of this? They are known to be somewhat out of tune at certain stages of the scale... are the holes lined up so that they have the same equal distances between the notes like a piano or is there something else going on? All I know is that when I play sax I have to change embouchure depending on where I am in term of how many holes are open/closed if I want to stay fully in tune with the band. But now I'm wondering if piano and sax are actually on the same page and for reasons mentioned in the video it's just not the same page as the rest of the band.
Anyone?
Tldr: none of the powers of the twelfth root of two are rational.
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Can someone give me a 'TL;DR', please?